Superstrings defined in \(D = 10 \quad \Rightarrow \quad \mathcal{M}^{1, 9} = \mathcal{M}^{1, 3} \otimes X_6\)
Requirements
- \(X_6\) is a compact manifold \((M, g)\)
- \(N = 1\) SUSY in 4D
- SM \(\subset\) arising gauge algebra
Solution
- \(\dim_{\mathbb{C}} M = 3\)
- \(\mathrm{Hol}(g) \subseteq \mathrm{SU}(3)\)
- \(\mathrm{Ric}(g) \equiv 0\) or \(c_1(M) \equiv 0\)
Calabi-Yau Manifolds
- no known metric for compact CY
- need to study topology (Hodge numbers) to infer 4D properties
\[h^{r,s} = \dim_{\mathbb{C}} \mathrm{H}_{\overline{\partial}}^{r,s}(M, \mathbb{C})\]
